Metric-based
approaches
to mechanics
allow one to accomplish a great deal in context of a
single reference frame of yardsticks and synchronized
clocks, before moving to considerations that involve more
than one such frame. One of these things is the quantitative
treatment of accelerated motion. This is especially clear
if one plots velocity-parameters derived from the metric
equation e.g. coordinate-velocity v=dx/dt,
proper-velocity
w=dx/dτ, and gamma γ=dt/dτ as a function of
map-position. Rather complex scenarios, like that of
an accelerated-twin adventure, can be illustrated in very
quantitative and concrete terms. Such plots
can be made dimensionless in terms of a fixed constant
proper-acceleration α. Since these apply locally to all
accelerated motions (even in curved spacetime), we
refer to them here as "universal acceleration plots".

In this context, below find an interactive plot of velocities
(green is coordinate-velocity or v=dx/dt;
blue is proper-velocity or w=dx/dτ)
and gamma
(red is speed-of-map-time or γ=dt/dτ=E/mc2)
for a system undergoing
constant proper-acceleration α, versus displacement
parallel and perpendicular to that acceleration.
All distances (and simultaneity) are defined with
respect to a single reference
"map-frame". The
velocity-axis is in the "facing funnel" direction,
while perpendicular displacements lie along the
"butterfly symmetry axis".
The section of the universal plot provided here covers
values of coordinate-velocity v (green) for map-times t between -2 and 2,
and of proper-velocity w (blue) for proper-times τ between -2 and 2,
with all times
measured relative to the "turn-around" event (vx=0)
in units of
c/α (or years at 1 gee). Values of transverse
coordinate-velocity (a constant for a given trajectory)
range from -0.9c to 0.9c (i.e. around 2 lightyears/traveler
year). The red surface
represents γ for x-distance
traveled relative to rest of 0 to +2, in units of
c2/α (or lightyears at 1 gee).

Here comes the applet...

On slower computers it may take some time
until the image appears. Dragging the mouse
across the image can rotate or spin
the plot, while dragging the mouse up or
down with s-down on your keyboard can
zoom in or out. Left click enables
many javaview pulldown dialogs.

See if you can determine, with help from your browser,
how deep into the funnel the red γ "cap" extends.

Universal acceleration plots are useful, for example, if one
wants to visualize
an
accelerated-twin adventure in concrete terms.
A (1+1)-dimensional example of this is shown below. Please excuse
the arcane "one-map three-clock" nomenclature, which in
addition to the above velocities plots map-time t and proper-time
τ in dimensionless form, as well as "chase-plane" time T
and "Galilean-kinematic" velocity V=dx/dT, for which the
chase-plane trajectory has been chosen so that
Galileo's constant acceleration equations describe the motion of
our accelerated traveler.
The most interesting times and velocities, of course, are
the ones mentioned above: map-time t and coordinate-velocity v=dx/dt,
as well as the less frame-variant quantitites: proper-time τ,
proper-velocity w=dx/dτ, and gamma γ=dt/dτ.

Note in the changing plot on the right that as the round trip duration
and maximum velocity decreases, the time-lapse disagreements between
traveler and map get smaller. As the trip becomes longer, traveler
time elapsed can become almost negligible in comparison to map time.